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Analyticity and Higher Twists in Hadron Structure: Resolving the Confinement Problem through QCD

This presentation delves into the complex processes and classifications of non-perturbative inputs in Quantum Chromodynamics (QCD), akin to biology before Darwin. It explores the compatibility of these classifications with symmetries, causality, and analyticity. Topics covered include HT resummation, scaling variables, TMDs, GPDs, TDs, and more.

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Analyticity and Higher Twists in Hadron Structure: Resolving the Confinement Problem through QCD

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  1. Analyticity and higher twists Hadron Structure’13, Tatranské Matliare, July 2, 2013 Oleg Teryaev JINR, Dubna

  2. QCD without confinement problem solved – description and classification of NP inputs (~biology before Darwin)Complicated processes – complicated classification – UGDFs, TMDs, GPDs, HTs (~multiparton pdfs) Are such classifications compatible with symmetries ( -> low-energy theorems), CAUSALITY AND ANALYTICITY ?

  3. Outline • HT resummation and analyticity in (spin-dependent) DIS • HT resummation and scaling variables: DIS vs SIDIS • TMDs as infinite towers of twists • Quarks in vacuum and inside the hadrons: TMDs vs non-local condensates

  4. Spin dependent DIS • Two invariant tensors • Only the one proportional to contributes for transverse (appears in Born approximation of PT) • Both contribute for longitudinal • Apperance of only for longitudinal case –result of the definition for coefficients to match the helicity formalism

  5. Generalized GDH sum rule • Define the integral – scales asymptotically as • At real photon limit (elastic contribution subtracted) – - Gerasimov-Drell-Hearn SR • Finite limit of infinite sum of inverse powers! • Proton- dramatic sign change at low Q2!

  6. Finite limit of infinite sum of inverse powers?! • How to sum ci (- M2/Q2 )i ?! • May be compared to standard twist 2 factorization • Light cone:

  7. Moments and partonic expression • Lorentz invariance: • Summed by representing

  8. Summation and analyticity • Justification (in addition to nice parton picture) - analyticity! • Correct analytic properties of virtual Compton amplitude • Defines the region of x • Generalized Parton Distributions – compatibility of analyticity with factorization – non-trivial: Radon transform technique (OT’05; Anikin,OT’07,D. Muller,Kumericky’09-13)

  9. Summation and analyticity-HT • Parton model with |x| < 1 – transforms poles to cuts! – justifies the representation in terms of moments • For HT series ci = <f(x) xi> - moments of HT “density”- geometric series rather than exponent: Σci (- M2/Q2 ) = < M2f(x)/(x M2+ Q2 )> • Like in parton model: pole -> cut

  10. Summation and analyticity-HT • Analytic properties  proper integration region (positive x, two-pion threshold) • Finite value for Q2 =0: -< f(x)/x> - inverse moment! • Relation to matrix elements unclear (probably – Wilson lines: transverse momentum)

  11. Summation and analyticity • “Chiral” expansion: - (- Q2/M2 )i <f(x)/x i+1> • “Duality” of chiral and HT expansions: analyticity allows for EITHER positive OR negative powers (no complete series!) • Analyticity – (typically)alternating series

  12. Summation and analyticity • Analyticity of HT  analyticity of pQCD series – (F)APT • Finite limit -> series starts from 1/Q2 unless the density oscillates • Annihilation – (unitarity - no oscillations) extra justification of “short strings”?

  13. Short strings • Confinement term in the heavy quarks potential – dimension 2 (GI OPE – 4!) scale ~ tachyonic gluon mass • Effective modification of gluon propagator

  14. Back to DIS: (J. Soffer, OT ‘92) • Supported by the fact that • Linear in , quadratic term from • Natural candidate for NP, like QCD SR analysis – hope to get low energy theorem via WI (C.f. pion F.F. – Radyushkin) - smooth model • For -strong Q – dependence due to Burkhardt-Cottingham SR

  15. Models for :proton • Simplest - linear extrapolation – PREDICTION (10 years prior to the data) of low (0.2 GeV) crossing point • Accurate JLAB data – require model account for PQCD/HT correction – matching of chiral and HT expansion • HT – values predicted from QCD SR (Balitsky, Braun, Kolesnichenko) • Rather close to the data For Proton

  16. The model for transition to small Q (Soffer, OT ’04)

  17. Access to the neutron – via the (p-n) difference – linear in -> Deuteron – refining the model eliminates the structures Models for :neutron and deuteron for neutron and deuteron

  18. Duality for GDH – resonance approach • Textbook (Ioffe, Lipatov. Khoze) explanation of proton GGDH structure –contribution of dominant magnetic transition form factor • Is it compatible with explanation?! • Yes!– magnetic transition contributes entirely to and as a result to

  19. Bjorken Sum Rule – most clean test • Strongly differs from smooth interpolation for g1 • Scaling down to 1 GeV

  20. New option: Analytic Perturbation Theory • Shirkov, Solovtsov: Effective coupling – analytic in Q2 • Landau pole automatically removed • Generic processes: F(ractional)APT • Does not include full NPQCD dynamics (appears at ~ 1GeV where coupling is still small) –> Higher Twist • Depend on (A)PT • Low Q – very accurate data from JLAB

  21. Bjorken Sum Rule-APT Accurate data + IR stable coupling -> low Q region

  22. PT/HT duality

  23. Matching in PT and APT • Duality of Q and 1/Q expansions

  24. 4-loop corrections includedV.L. Khandramai, R.S. Pasechnik, D.V. Shirkov, O.P. Solovtsova, O.V. Teryaev. Jun 2011. 6 pp. e-Print: arXiv:1106.6352 [hep-ph] • HT decrease with PT order and becomes compatible to zero (V.I. Zakharov’s duality) • Analog for TMD – intrinsic/extrinsic TM duality!?

  25. Asymptotic series and HT • Duality: HT can be eliminated at all (?!) • May reappear for asymptotic series - the contribution which cannot be described by series due to its asymptotic nature.

  26. Another version of IR stable coupling – “gluon mass” – Cornwall,.. Simonov,.. Shirkov(NLO)arXiv:1208.2103v2 [hep-th] 23 Nov 2012 • HT – in the “VDM” form M2/(M2+ Q2 ) • Corresponds to f(x) ~ • Possible in principle to go to arbitrarily small Q • BUT NO matching with GDH achieved • Too large average slope – signal for transverse polarization role !

  27. Account for transverse polarization -> description in the whole Q region (Khandramai, Shirkov, OT, in progress) • 1-st order – LO coupling with (P) gluon mass + (NP) “VDM” • GDH – relation between P and NP masses

  28. P/NP

  29. Data vs LO/NLO/NNLO

  30. Smaller Q

  31. HT – modifications of scaling variables • Various options since Nachtmann • ~ Gluon mass • -//- new (spectrality respecting) modification • JLD representation

  32. Resummed twists: Q->0 (D. Kotlorz, OT)

  33. Modified scaling variable for TMD • First appeared in P. Zavada model • XZ = • Suggestion – also (partial) HT resummation(M goes from denominator to numerator in cordinate/impact parameter space)?!

  34. HT for TMDs - case study: Collins FF and twist 3 • x(T) –space : qq correlator ~ M - twist 3 • Cf to momentum space (kT/M) – M in denominator – “Leading Twist” • x <-> kT spaces • Moment – twist 3 (for Sivers – Boer, Mulders, Pijlman) • Higher (2D-> Bessel) moments – infinite tower of twists (for Sivers - Ratcliffe,OT)

  35. Resummation in x-space (DY) • Full x/kT – dependence and its expansion • Singularities (-> power/log tail – cf Efremov, Vladimirov – causality -  arXiv:1306.3929 ) should be subtracted to get exponential falloff (required to have all moments in TM finite) • DY weighted cross-section

  36. What happens in vacuum? • Suggestion : similarity with non-local quark condensates (Radyushkin et al) : quarks in vacuum ~ transverse d.o.f. of quarks in hadrons (Euclidian!) ?! • Universality in hadron( type-dependent) TMD and vacuum NLC functions?!

  37. Hadronic vs vacuum matrix elements • Hadron -> (LC) momentum; dimension-> twist • Transverse dynamics – looks more similar to vacuum: quark virtuality -> TM(squared) • (Euclidian) space separation -> impact parameter

  38. More complcated objects • Modification of BFKL kernel - talk of E. Levin: May be a way to modify NP part (impact factor): exponential falloff in coordinate space -> finiteness in momentum space (cf GDH) • D-term for GPDs (~ quadrupole gravitational FF) ~ Cosmological Constant in vacuum; Negative D-term -> negative CC in space-like/positive in time-like regions: Annihilation~Inflation

  39. Conclusions • Representation for HT similar to parton model: preserves analyticity changing the poles to cuts • Infinite sums of twists – important for DIS at Q->0 • Good description of the data at all Q2 with the single scale parameter

  40. Discussion • TMD – infinite towers of twists • Modified scaling variables – models for infinite twists towers in DIS and SIDIS • Similar to non-local quark condensates – vacuum/hadrons universality?!

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